Abstract
We show that an upper bound function for the Church–Rosser theorem of type-free λ-calculus with β-reduction must be in the fourth level of the Grzegorczyk hierarchy, i.e., the smallest Grzegorczyk class properly extending the class of elementary functions. At this level we also find common reducts for the confluence property. The proof method here can be applied not only to type-free λ-calculus with βη-reduction but also to typed λ-calculi such as Pure Type Systems.
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